Members MadKeithV Posted October 12, 2009 Members Share Posted October 12, 2009 .999_ is less than 1 by an infinitely small difference, but a difference nonetheless. No, it isn't smaller than 1. Several proofs have been posted already. See also 0.9 Recurring, in Wikipedia. The misconception usually seems to stem from the idea that the decimal system has only one representation for a number. There's no question that 1.0 is the *preferred* representation, but 0.9 recurring is the same number with a different representation. Another way to look at it: there is no "infinitely small" real number that you could add to 0.9 recurring to get one. If you have any small number greater than zero, then there is a smaller number between that small number and zero as well. The real numbers are continuous. This is also known as the intermediate value theorem. Quote Link to comment Share on other sites More sharing options...
Members zack Posted October 12, 2009 Members Share Posted October 12, 2009 And how much is that in pubes? Quote Link to comment Share on other sites More sharing options...
Members zack Posted October 12, 2009 Members Share Posted October 12, 2009 Could the difference between 1 and .9_ be written as .1x10^-? Quote Link to comment Share on other sites More sharing options...
Members Digital Jams Posted October 12, 2009 Members Share Posted October 12, 2009 Probably one of the better threads imo, people are thinking more in this thread than any other one currently. That is a parlor trick equation. Quote Link to comment Share on other sites More sharing options...
Members MadKeithV Posted October 12, 2009 Members Share Posted October 12, 2009 Could the difference between 1 and .9_ be written as .1x10^-? Probably yes, but why bother if 0 is much shorter. Quote Link to comment Share on other sites More sharing options...
Members Jesse G Posted October 12, 2009 Members Share Posted October 12, 2009 No, it isn't smaller than 1. Several proofs have been posted already. See also 0.9 Recurring, in Wikipedia. The misconception usually seems to stem from the idea that the decimal system has only one representation for a number. There's no question that 1.0 is the *preferred* representation, but 0.9 recurring is the same number with a different representation. Another way to look at it: there is no "infinitely small" real number that you could add to 0.9 recurring to get one. If you have any small number greater than zero, then there is a smaller number between that small number and zero as well. The real numbers are continuous. This is also known as the intermediate value theorem. Well goddamn. Quote Link to comment Share on other sites More sharing options...
Members MadKeithV Posted October 12, 2009 Members Share Posted October 12, 2009 Well goddamn. Once you get into infinite-dimensional integrals, things get a bit more hairy though. Quote Link to comment Share on other sites More sharing options...
Members LP50 Posted October 13, 2009 Members Share Posted October 13, 2009 I understand the logic, but by my calculations 3/2 is 2, not 1. Because you would round up removing the decimal places. I believe your teachers logic is flawed as well as his math. Integer Division. There is no rounding. Quote Link to comment Share on other sites More sharing options...
Members Corey112 Posted October 13, 2009 Members Share Posted October 13, 2009 Asked my college calculus professor about this today, and he said that _ .9 does equal 1. He explained it this way. 1/9 = .1 repeating... 2/9 = .2 repeating 3/9 = .3 repeating 4/9 = .4 repeating 5/9 = .5 repeating etc etc.... 8/9 = .8 repeating 9/9 = 1. Whether that is the actual proof Quote Link to comment Share on other sites More sharing options...
Members guitarcapo Posted October 13, 2009 Members Share Posted October 13, 2009 If there are an infinite number of points possible between two points...how can someone walk over an infinite number of points? (one of Xeno's paradoxes) hint: "infinite number?" Quote Link to comment Share on other sites More sharing options...
Members sdconvoy Posted October 13, 2009 Members Share Posted October 13, 2009 why are there people saying that .999... APPROACHES 1... it's not like .9(repeating) is a function... it's a number... besides, if 1 == .999... , then there would be some kind of difference, or a number you could add to .999... to bring it to one. no one can describe any actual number to fit here, so they are equivalent. Quote Link to comment Share on other sites More sharing options...
Members Captain Commie Posted October 13, 2009 Members Share Posted October 13, 2009 If there are an infinite number of points possible between two points...how can someone walk over an infinite number of points? (one of Xeno's paradoxes) hint: "infinite number?" because all points are an element of the interval you have defined. Quote Link to comment Share on other sites More sharing options...
Members MadKeithV Posted October 13, 2009 Members Share Posted October 13, 2009 If there are an infinite number of points possible between two points...how can someone walk over an infinite number of points? (one of Xeno's paradoxes) hint: "infinite number?" There's a fundamental wrong assumption in that paradox, assuming that a mathematical infinity maps onto reality. Reality seems to be discrete, i.e. if you break it down far enough you end up with the "smallest possible size" - see Planck Length. Quote Link to comment Share on other sites More sharing options...
Members nvranka Posted October 13, 2009 Members Share Posted October 13, 2009 Quoted for fail. how can u even try to quote others for fail when your OP was so terribly terribly wrong? Quote Link to comment Share on other sites More sharing options...
Members inkblot Posted October 13, 2009 Members Share Posted October 13, 2009 There's a fundamental wrong assumption in that paradox, assuming that a mathematical infinity maps onto reality. Reality seems to be discrete, i.e. if you break it down far enough you end up with the "smallest possible size" - see Planck Length. I would argue that the fundamentally wrong assumption is that an infinite sum of lengths must result in infinite length. If this were true, it would be impossible to have asymptotic behavior, which in reality is readily apparent in simple mathematical functions. The geometric series describes Zeno's motion paradox. In case anyone wants to argue that the sum of the series only converges to 1 and never reaches it, therefore Zeno was right, should note that even if you don't believe the series reaches 1, it does get bigger than zero - which is where Zeno thought it was stuck. It would seem obvious that if you can travel 99.999999% of the distance you can travel that last bit too Quote Link to comment Share on other sites More sharing options...
Members thinkpad20 Posted October 13, 2009 Members Share Posted October 13, 2009 The problem here is that many people don't understand what infinity means, mathematically. Which isn't surprising; it's a tough concept, especially seeing as infinity (arguably) doesn't really exist in the world that we experience. But mathematically, it is very real and very well understood. Quote Link to comment Share on other sites More sharing options...
Members Robson780 Posted October 13, 2009 Members Share Posted October 13, 2009 A+ thread would read again Quote Link to comment Share on other sites More sharing options...
Members MadKeithV Posted October 13, 2009 Members Share Posted October 13, 2009 The problem here is that many people don't understand what infinity means, mathematically. Which isn't surprising; it's a tough concept, especially seeing as infinity (arguably) doesn't really exist in the world that we experience. But mathematically, it is very real and very well understood. Aleph null, or aleph one infinity? ;-) Quote Link to comment Share on other sites More sharing options...
Members Shenaniganizer Posted October 13, 2009 Members Share Posted October 13, 2009 Hi guys. Quote Link to comment Share on other sites More sharing options...
Members thinkpad20 Posted October 13, 2009 Members Share Posted October 13, 2009 Aleph null, or aleph one infinity? ;-) Unfortunately, I never really went that far in set theory... I was a physics major so most of my math was based on calculus and a little probability. But I had a good dose of (introductory) number theory along the way. Quote Link to comment Share on other sites More sharing options...
Members MadKeithV Posted October 13, 2009 Members Share Posted October 13, 2009 Unfortunately, I never really went that far in set theory... I was a physics major so most of my math was based on calculus and a little probability. But I had a good dose of (introductory) number theory along the way. Aleph null infinity is "countable" - like the set of integers. They are infinite, but there is only a finite number of integers in any interval. Aleph one infinity is not countable - like the set of real numbers. Any interval of the real numbers has an infinite number of real numbers. Quote Link to comment Share on other sites More sharing options...
Members thinkpad20 Posted October 13, 2009 Members Share Posted October 13, 2009 Aha. Yeah, I know countable and uncountable... didn't know that terminology though. Quote Link to comment Share on other sites More sharing options...
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